This application claims benefit of Japanese Application No. 2000-234056 filed in Japan on Aug. 2, 2000, the contents of which are incorporated by this reference.
1. Field of the Invention
The present invention relates to an optical element using an organic-inorganic composite material. More particularly, the present invention relates to an optical element using organic-inorganic composite material which is suitable for use in optical systems such as an image pickup system and an ocular system.
2. Discussion of Related Art
Conventional transparent optical elements are produced by using a glass material (inorganic material) or a plastic material (organic material). When a glass material is used to produce an optical element, polishing is generally performed to form the material into the desired shape. Therefore, a great deal of time is required to carry out the polishing process. In the case of a plastic material, on the other hand, an optical element can be produced by injection molding. Accordingly, optical elements formed from a plastic material are superior in productivity but inferior in temperature characteristics. In addition, the number of kinds of plastic materials usable to produce optical elements is unfavorably small. In particular, there is no plastic material exhibiting a high refractive index.
There has recently been known a method of producing an optical element from a glass material by molding. With this method, however, the kinds of glass materials usable for production by molding are limited. Usable glass materials are restricted within a very narrow range. There is also a method wherein an optical element is molded from a thermosetting resin material as a plastic material. This method also has the disadvantage that usable thermosetting resin materials are limited. In particular, there is no thermosetting resin material having a high refractive index and exhibiting minimal dispersion.
Let us put the above-described problems in order.
Problems of glass materials:
1. Heavy in specific gravity.
2. Inferior in productivity.
Problems of plastic materials:
1. There is no plastic material having a high refractive index and exhibiting minimal dispersion.
2. Plastic materials are hygroscopic.
3. Exhibiting birefringence.
4. Weak in mechanical strength
Conventional optical elements using a glass material are heavy in specific gravity. Therefore, an increase in weight gives rise to a problem, particularly in the case of an optical element having a large lens aperture, e.g. an ocular lens. Weight is a matter of great concern for an optical element having a large aperture, e.g. a telephotographic lens for use with a camera. In addition, because lenses using a glass material are produced by polishing, much time is required for the polishing process.
Regarding plastic materials, there is no plastic material having a high refractive index, in particular. High-refractive index materials used for eyeglasses exhibit large dispersion and hence cause large chromatic aberrations. Because they are thermosetting plastic materials, a great deal of time is required for setting. Further, because plastic materials are hygroscopic, optical elements molded therefrom absorb water contained in the air even under normal environmental conditions, causing a change in shape and also a change in refractive index.
The present invention was made in view of the above-described problems with the prior art.
An object of the present invention is to provide a lightweight and homogeneous optical element exhibiting favorably weak birefringence and hygroscopicity as well as superior productivity and producing minimal chromatic aberrations by using an organic-inorganic composite material having both the properties of a glass material and those of a plastic material.
To attain the above-described object, the present invention provides an optical element having at least one entrance refracting surface and at least one exit refracting surface. The optical element is formed from an organic-inorganic composite material.
The optical element using an organic-inorganic composite material according to the present invention will be described below.
Recently, attention has being paid to organic-inorganic composite materials having an ultramicro structure in which an inorganic phase is dispersed in the three-dimensional network (matrix) of an organic phase as organic-inorganic hybrid materials [for example, see the August 1998 issue of xe2x80x9cEngineering Materialsxe2x80x9d (Vol. 16, No. 8), pp. 26-31, and the September 1999 issue of xe2x80x9cMaterials Sciencexe2x80x9d (Vol. 36, No. 5, pp. 39-45]. Such an organic-inorganic hybrid material has both the characteristic features of an organic polymer (e.g. moldability and low density) and those of an inorganic compound (e.g. transparency). The physical properties of the organic-inorganic hybrid material conform to the rules of composition of the organic and inorganic components.
Accordingly, it is possible to obtain a material of high refractive index and low dispersion, which cannot be realized with a plastic material, by appropriately selecting a material for the inorganic phase and a material for the organic phase and selecting an appropriate mixture ratio for the two materials. The organic-inorganic hybrid material thus obtained exhibits superior moldability and hence allows an optical element of desired shape to be obtained by injection molding.
Such an organic-inorganic hybrid material may be produced, for example, by mixing together an organic polymer and a metal alkoxide or a glass precursor (e.g. tetraethoxy silane). Organic-inorganic composite materials obtained in this way include one in which oxygen in the organic polymer and the protons of Sixe2x80x94OH groups in the inorganic polymer bond to each other by hydrogen bonding. Another organic-inorganic composite material has chemical bonding between the organic polymer and the inorganic disperse phase. For example, there is an organic-inorganic composite material having covalent bonding such as xe2x80x94NHxe2x80x94NCOHxe2x80x94 as a result of polymerization of xe2x80x94NH2 group at the end of the inorganic disperse phase with xe2x80x94NCO group.
To ensure the transparency of such an organic-inorganic composite material, it is necessary that the micro structure of the dispersed inorganic phase should have a size not more than the working wavelength. It is desirable that the size of the micro structure should be not larger than 200 nanometers, preferably not larger than 100 nanometers, even more preferably not larger than 20 nanometers.
As the inorganic component of an organic-inorganic composite material, it is preferable to use one selected from among inorganic matters (oxides and nitrides) containing metals belonging to Group IVA in the periodic table of the elements, e.g. Ti and Zr, metals belonging to Group IIIB, e.g. Al, and metals belonging to Group IVB, e.g. Si and Ge. From the viewpoint of ease of use, Si, Ti, Al and Zr are particularly preferable. As a component for imparting high-refractive index properties, it is preferable to use a metal belonging to Group IIIA, e.g. a lanthanoid such as Y or La, or a metal belonging to Group VA, e.g. Nb or Ta, in the form of a composite material combined with Si, Ti, Al, or Zr. It is possible to use metal alkoxides containing these components, derivatives and metal salts thereof, etc.
As the organic component of an organic-inorganic composite material, it is possible to use most organic polymers compatible with the inorganic material used in the process of mixing, gelation, drying and setting and capable of forming hydrogen bonds or covalent bonds with the inorganic material. More specifically, substances containing structures such as xe2x80x94COOH group, xe2x80x94NH2 group, xe2x80x94OH group and a group containing S have interaction with xe2x80x94H, COOH group, xe2x80x94NH2 group, xe2x80x94OH group, etc. contained in oligomers formed by hydrolysis of a part of metal alkoxides or derivatives used as inorganic matter. Various organic matters are usable as materials that form hydrogen bonds or covalent bonds with the inorganic material, e.g. polyurethane, urea resin, polyamide, polyimide, polycarbonate, and polyvinyl alcohol. To form covalent bonds with the inorganic material, the above-described organic matter is used after a functional group such as an alkoxide group capable of reacting with the inorganic material has been introduced into the organic polymer chain. Generally speaking, organic-inorganic hybrid materials in which the organic material covalently bonds to the inorganic material are superior in mechanical strength because covalent bonding is stronger than hydrogen bonding.
Incidentally, when an optical element is formed by using an organic-inorganic composite material as stated above, it is desirable that at least two transmitting surfaces of the optical element should have an optical power.
If a transmitting surface of an optical element has an optical power, chromatic aberrations produced by the refracting surface increase. The increased chromatic aberrations of the optical element can be corrected by another optical element when an optical system is constructed by combining together a plurality of optical elements. In this regard, it is preferable to use the above-described organic-inorganic composite material. With the organic-inorganic composite material, it is possible to form an optical element of low dispersion and hence possible to construct an optical system using an optical element producing minimal chromatic aberrations.
In addition, the use of the above-described organic-inorganic composite material allows the refractive index to be increased. Consequently, it becomes possible to minimize spherical aberration, astigmatism and image distortion. The power P of a surface is expressed by P=(nxe2x88x921)/R (n is the refractive index and R is the radius of curvature). Accordingly, the curvature radius R can be increased according as the refractive index n becomes higher. Thus, it is possible to minimize aberrations produced by the optical element.
In addition, the optical element may have at least one reflecting surface. When a light ray passing through the center of an object and the center of a stop is defined as an axial principal ray, the optical element may be arranged so that the axial principal ray is bent in the organic-inorganic composite material.
If the optical element has one reflecting surface, the optical path of the axial principal ray passes through the same material twice. Consequently, the optical element needs a sufficiently high transmittance. A favorably high transmittance can be obtained by using the above-described organic-inorganic composite material, which is less birefringent and hygroscopic than plastic materials. Further, if a reflecting surface is included in the optical element, the axial principal ray passes along a folded optical path, resulting in an increase in the optical path length. Accordingly, the volumetric capacity of the optical element can be reduced in comparison to a lens having only refracting surfaces. Therefore, it is even more preferable from the viewpoint of achieving a reduction in weight that the optical element should have a reflecting surface.
In this case, it is desirable that the at least one reflecting surface should have an optical power.
A reflecting surface having an optical power is usually constructed in the form of a back-coated mirror. Therefore, the amount of aberrations produced in the optical element reduces favorably. The optical power of a back-coated mirror is expressed by P=2n/R. Accordingly, the effect of using a high refractive index becomes more remarkable than in the case of an optical element formed from only transmitting surfaces. In addition, it is possible to reduce the power assigned to the transmitting surfaces by an amount corresponding to the power given to the reflecting surface. Consequently, the amount of aberrations produced in the optical element can be minimized synergistically by the effect of using an reflecting surface with an optical power and the effect of using a high refractive index.
It is desirable that the at least one reflecting surface should have a rotationally asymmetric surface configuration that corrects decentration aberrations due to decentration.
When the optical path is bent by a reflecting surface, decentration aberrations occur to a considerable extent. It is difficult to correct the decentration aberrations by using the transmitting surfaces. The decentration aberrations can be corrected by a rotationally asymmetric surface. Therefore, a rotationally asymmetric surface configuration is given to the reflecting surface to correct the decentration aberrations.
In the present invention, a free-form surface is used as a typical example of a surface having a rotationally asymmetric curved surface configuration. A free-form surface is defined by the following equation. The Z-axis of the defining equation is the axis of a free-form surface.                     Z        =                                            cr              2                        /                          [                              1                +                                                      {                                          1                      -                                                                        (                                                      1                            +                            k                                                    )                                                ⁢                                                  c                          2                                                ⁢                                                  r                          2                                                                                      }                                                              ]                                +                                    ∑                              j                =                2                            66                        ⁢                                          C                j                            ⁢                              X                m                            ⁢                              Y                n                                                                        (        a        )            
In the equation (a), the first term is a spherical surface term, and the second term is a free-form surface term.
In the spherical surface term:
c: the curvature at the vertex
k: a conic constant
r=√{square root over ( )}(X2+Y2)
The free-form surface term is given by                                           ∑                          j              =              2                        66                    ⁢                                    C              j                        ⁢                          X              m                        ⁢                          Y              n                                      =                ⁢                                            C              2                        ⁢            X                    +                                    C              3                        ⁢            Y                    +                                                ⁢                                            C              4                        ⁢                          X              2                                +                                    C              5                        ⁢            X            ⁢                          xe2x80x83                        ⁢            Y                    +                                    C              6                        ⁢                          Y              2                                +                                                ⁢                                            C              7                        ⁢                          X              3                                +                                    C              8                        ⁢                          X              2                        ⁢            Y                    +                                    C              9                        ⁢            X            ⁢                          xe2x80x83                        ⁢                          Y              2                                +                                    C              10                        ⁢                          Y              3                                +                                                ⁢                                            C              11                        ⁢                          X              4                                +                                    C              12                        ⁢                          X              3                        ⁢            Y                    +                                    C              13                        ⁢                          X              2                        ⁢                          Y              2                                +                                    C              14                        ⁢            X            ⁢                          xe2x80x83                        ⁢                          Y              3                                +                                    C              15                        ⁢                          Y              4                                +                                                ⁢                                            C              16                        ⁢                          X              5                                +                                    C              17                        ⁢                          X              4                        ⁢            Y                    +                                    C              18                        ⁢                          X              3                        ⁢                          Y              2                                +                                    C              19                        ⁢                          X              2                        ⁢                          Y              3                                +                                    C              20                        ⁢            X            ⁢                          xe2x80x83                        ⁢                          Y              4                                +                                                ⁢                                            C              21                        ⁢                          Y              5                                +                                                ⁢                                            C              22                        ⁢                          X              6                                +                                    C              23                        ⁢                          X              5                        ⁢            Y                    +                                    C              24                        ⁢                          X              4                        ⁢                          Y              2                                +                                    C              25                        ⁢                          X              3                        ⁢                          Y              3                                +                                    C              26                        ⁢                          X              2                        ⁢                          Y              4                                +                                                ⁢                                            C              27                        ⁢            X            ⁢                          xe2x80x83                        ⁢                          Y              5                                +                                    C              28                        ⁢                          Y              6                                +                                                ⁢                                            C              29                        ⁢                          X              7                                +                                    C              30                        ⁢                          X              6                        ⁢            Y                    +                                    C              31                        ⁢                          X              5                        ⁢                          Y              2                                +                                    C              32                        ⁢                          X              4                        ⁢                          Y              3                                +                                    C              33                        ⁢                          X              3                        ⁢                          Y              4                                +                                                ⁢                                            C              34                        ⁢                          X              2                        ⁢                          Y              5                                +                                    C              35                        ⁢            X            ⁢                          xe2x80x83                        ⁢                          Y              6                                +                                    C              36                        ⁢                          Y              7                                                                      ⁢        …            
where Cj (j is an integer of 2 or higher) are coefficients.
In general, the above-described free-form surface does not have planes of symmetry in both the XZ- and YZ-planes. However, a free-form surface having only one plane of symmetry parallel to the YZ-plane is obtained by making all terms of odd-numbered degrees with respect to X zero. A free-form surface having only one plane of symmetry parallel to the XZ-plane is obtained by making all terms of odd-numbered degrees with respect to Y zero.
In addition, free-form surfaces as the above-described surfaces with a rotationally asymmetric curved surface configuration may be defined by Zernike polynomials. That is, the configuration of a free-form surface may be defined by the following equation (b). The Z-axis of the defining equation (b) is the axis of Zernike polynomial. A rotationally asymmetric surface is defined by polar coordinates of the height of the Z-axis with respect to the XY-plane. In the equation (b), R is the distance from the Z-axis in the XY-plane, and A is the azimuth angle about the Z-axis, which is expressed by the angle of rotation measured from the Z-axis.                                                         x              =                            ⁢                              R                xc3x97                cos                ⁢                                  xe2x80x83                                ⁢                                  (                  A                  )                                                                                                        y              =                            ⁢                              R                xc3x97                sin                ⁢                                  xe2x80x83                                ⁢                                  (                  A                  )                                                                                                        z              =                            ⁢                                                D                  2                                +                                                      D                    3                                    ⁢                  R                  ⁢                                      xe2x80x83                                    ⁢                                      cos                    ⁡                                          (                      A                      )                                                                      +                                                      D                    4                                    ⁢                  R                  ⁢                                      xe2x80x83                                    ⁢                                      sin                    ⁡                                          (                      A                      )                                                                      +                                                      D                    5                                    ⁢                                      R                    2                                    ⁢                                      cos                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                      D                    6                                    ⁡                                      (                                                                  R                        2                                            -                      1                                        )                                                  +                                                                                                      ⁢                                                                    D                    7                                    ⁢                                      R                    2                                    ⁢                                      sin                    ⁡                                          (                                              2                        ⁢                        A                                            )                                                                      +                                                      D                    8                                    ⁢                                      R                    3                                    ⁢                                      cos                    ⁡                                          (                                              3                        ⁢                        A                                            )                                                                      +                                                                            D                      9                                        ⁡                                          (                                                                        3                          ⁢                                                      R                            3                                                                          -                                                  2                          ⁢                          R                                                                    )                                                        ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                    A                    )                                                  +                                                                                                      ⁢                                                                                          D                      10                                        ⁡                                          (                                                                        3                          ⁢                                                      R                            3                                                                          -                                                  2                          ⁢                          R                                                                    )                                                        ⁢                                      sin                    ⁡                                          (                      A                      )                                                                      +                                                      D                    11                                    ⁢                                      R                    3                                    ⁢                                      sin                    ⁡                                          (                                              3                        ⁢                        A                                            )                                                                      +                                                      D                    12                                    ⁢                                      R                    4                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          4                      ⁢                      A                                        )                                                  +                                                                                                      ⁢                                                                                          D                      13                                        ⁡                                          (                                                                        4                          ⁢                                                      R                            4                                                                          -                                                  3                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                      A                                        )                                                  +                                                      D                    14                                    ⁡                                      (                                                                  6                        ⁢                                                  R                          4                                                                    -                                              6                        ⁢                                                  R                          2                                                                    +                      1                                        )                                                  +                                                                                                      ⁢                                                                                          D                      15                                        ⁡                                          (                                                                        4                          ⁢                                                      R                            4                                                                          -                                                  3                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      xe2x80x83                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                      A                                        )                                                  +                                                      D                    16                                    ⁢                                      R                    4                                    ⁢                                      sin                    ⁡                                          (                                              4                        ⁢                        A                                            )                                                                      +                                                      D                    17                                    ⁢                                      R                    5                                    ⁢                                      cos                    ⁡                                          (                                              5                        ⁢                        A                                            )                                                                      +                                                                                                      ⁢                                                                                          D                      18                                        ⁡                                          (                                                                        5                          ⁢                                                      R                            5                                                                          -                                                  4                          ⁢                                                      R                            3                                                                                              )                                                        ⁢                                      cos                    ⁡                                          (                                              3                        ⁢                        A                                            )                                                                      +                                                                            D                      19                                        ⁡                                          (                                                                        10                          ⁢                                                      R                            5                                                                          -                                                  12                          ⁢                                                      R                            3                                                                          +                                                  3                          ⁢                          R                                                                    )                                                        ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                      A                                        )                                                  +                                                                                                      ⁢                                                                                          D                      20                                        ⁡                                          (                                                                        10                          ⁢                                                      R                            5                                                                          -                                                  12                          ⁢                                                      R                            3                                                                          +                                                  3                          ⁢                          R                                                                    )                                                        ⁢                                      xe2x80x83                                    ⁢                                      sin                    ⁡                                          (                      A                      )                                                                      +                                                                            D                      21                                        ⁡                                          (                                                                        5                          ⁢                                                      R                            5                                                                          -                                                  4                          ⁢                                                      R                            3                                                                                              )                                                        ⁢                                      xe2x80x83                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          3                      ⁢                      A                                        )                                                  +                                                                                                      ⁢                                                                    D                    22                                    ⁢                                      R                    5                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          5                      ⁢                      A                                        )                                                  +                                                      D                    23                                    ⁢                                      R                    6                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          6                      ⁢                      A                                        )                                                  +                                                                            D                      24                                        ⁡                                          (                                                                        6                          ⁢                                                      R                            6                                                                          -                                                  5                          ⁢                                                      R                            4                                                                                              )                                                        ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          4                      ⁢                      A                                        )                                                  +                                                                                                      ⁢                                                                                          D                      25                                        ⁡                                          (                                                                        15                          ⁢                                                      R                            6                                                                          -                                                  20                          ⁢                                                      R                            4                                                                          +                                                  6                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                      A                                        )                                                  +                                                      D                    26                                    ⁡                                      (                                                                  20                        ⁢                                                  R                          6                                                                    -                                              30                        ⁢                                                  R                          4                                                                    +                                              12                        ⁢                                                  R                          2                                                                    -                      1                                        )                                                  +                                                                                                      ⁢                                                                                          D                      27                                        ⁡                                          (                                                                        15                          ⁢                                                      R                            6                                                                          -                                                  20                          ⁢                                                      R                            4                                                                          +                                                  6                          ⁢                                                      R                            2                                                                                              )                                                        ⁢                                      xe2x80x83                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                      A                                        )                                                  +                                                                            D                      28                                        ⁡                                          (                                                                        6                          ⁢                                                      R                            6                                                                          -                                                  5                          ⁢                                                      R                            4                                                                                              )                                                        ⁢                                      xe2x80x83                                    ⁢                  sin                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          4                      ⁢                      A                                        )                                                  +                                                                                                      ⁢                                                D                  29                                ⁢                                  R                  6                                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  (                                      6                    ⁢                    A                                    )                                ⁢                …                                                                        (        b        )            
where Dm (m is an integer of 2 or higher) are coefficients. It should be noted that to design an optical system symmetric with respect to the X-axis direction, D4, D5, D6, D10, D11, D12, D13, D14, D20, D21, D22 . . . should be used.
The above defining equations are shown to exemplify surfaces with a rotationally asymmetric curved surface configuration. Therefore, the same advantageous effects can be obtained for any other defining equation that expresses such a rotationally asymmetric curved surface configuration.
It should be noted that other examples of defining equations for free-form surfaces include the following defining equation (c):
Z=xcexa3xcexa3CnmXY
Assuming that k=7 (polynomial of degree 7), for example, a free-form surface is expressed by an expanded form of the above equation as follows:                                                         Z              =                            ⁢                                                C                  2                                +                                                      C                    3                                    ⁢                  Y                                +                                                      C                    4                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    5                                    ⁢                                      Y                    2                                                  +                                                      C                    6                                    ⁢                  Y                  ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    7                                    ⁢                                      X                    2                                                  +                                                                                                      ⁢                                                                    C                    8                                    ⁢                                      Y                    3                                                  +                                                      C                    9                                    ⁢                                      Y                    2                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    10                                    ⁢                  Y                  ⁢                                      xe2x80x83                                    ⁢                                      X                    2                                                  +                                                      C                    11                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                                                                      ⁢                                                                    C                    12                                    ⁢                                      Y                    4                                                  +                                                      C                    13                                    ⁢                                      Y                    3                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    14                                    ⁢                                      Y                    2                                    ⁢                                      X                    2                                                  +                                                      C                    15                                    ⁢                  Y                  ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                      C                    16                                    ⁢                                      X                    4                                                  +                                                                                                      ⁢                                                                    C                    17                                    ⁢                                      Y                    5                                                  +                                                      C                    18                                    ⁢                                      Y                    4                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    19                                    ⁢                                      Y                    3                                    ⁢                                      X                    2                                                  +                                                      C                    20                                    ⁢                                      Y                    2                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                      C                    21                                    ⁢                  Y                  ⁢                                      xe2x80x83                                    ⁢                                      X                    4                                                  +                                                                                                      ⁢                                                                    C                    22                                    ⁢                                      "LeftBracketingBar"                                          X                      5                                        "RightBracketingBar"                                                  +                                                      C                    23                                    ⁢                                      Y                    6                                                  +                                                      C                    24                                    ⁢                                      Y                    5                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                      C                    25                                    ⁢                                      Y                    4                                    ⁢                                      X                    2                                                  +                                                      C                    26                                    ⁢                                      Y                    3                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                                                                      ⁢                                                                    C                    27                                    ⁢                                      Y                    2                                    ⁢                                      X                    4                                                  +                                                      C                    28                                    ⁢                  Y                  ⁢                                      "LeftBracketingBar"                                          X                      5                                        "RightBracketingBar"                                                  +                                                      C                    29                                    ⁢                                      X                    6                                                  +                                                      C                    30                                    ⁢                                      Y                    7                                                  +                                                      C                    31                                    ⁢                                      Y                    6                                    ⁢                                      "LeftBracketingBar"                    X                    "RightBracketingBar"                                                  +                                                                                                      ⁢                                                                    C                    32                                    ⁢                                      Y                    5                                    ⁢                                      X                    2                                                  +                                                      C                    33                                    ⁢                                      Y                    4                                    ⁢                                      "LeftBracketingBar"                                          X                      3                                        "RightBracketingBar"                                                  +                                                      C                    34                                    ⁢                                      Y                    3                                    ⁢                                      X                    4                                                  +                                                      C                    35                                    ⁢                                      Y                    2                                    ⁢                                      "LeftBracketingBar"                                          X                      5                                        "RightBracketingBar"                                                  +                                                                                                      ⁢                                                                    C                    36                                    ⁢                  Y                  ⁢                                      xe2x80x83                                    ⁢                                      X                    6                                                  +                                                      C                    37                                    ⁢                                      "LeftBracketingBar"                                          X                      7                                        "RightBracketingBar"                                    ⁢                  …                                                                                        (        c        )            
It should be noted that an anamorphic surface or a toric surface is also usable as a surface having a rotationally asymmetric curved surface configuration.
Further, it is desirable that the optical element according to the present invention should have at least two reflecting surfaces, and at least one of the at least two reflecting surfaces should have a rotationally asymmetric surface configuration that corrects decentration aberrations due to decentration.
If the optical element has at least two reflecting surfaces as stated above, the power is dispersed effectively, so that the aberrations can be corrected favorably.
The arrangement may be such that the optical element has two reflecting surfaces, and the entrance surface and the exit surface are disposed to face each other so that the axial principal ray passes along a substantially round-trip optical path.
If the entrance surface and the exit surface are disposed to face each other, the optical path becomes a substantially Z-shaped round-trip optical path. Consequently, it is possible to place the entrance surface and one reflecting surface side by side and it is also possible to place the exit surface and another reflecting surface side by side. Accordingly, a compact optical element can be constructed. The use of the organic-inorganic composite material allows the optical element to be further reduced in weight.
The entrance surface and the exit surface may be placed adjacent to each other so that the axial principal ray passes along a substantially intersecting optical path.
In this case, the optical path crosses itself within the optical element. The optical path passes through the same portion twice in different directions of passage of rays. Therefore, it is particularly important to use the organic-inorganic composite material, which exhibits favorably weak birefringence and hygroscopicity.
The optical element according to the present invention may be positioned in the vicinity of a stop of an optical system.
With the above-described arrangement, when the refractive index is increased, it becomes possible to suppress the occurrence of spherical aberration in particular. When the Abbe""s number is increased (i.e. dispersion is reduced), the occurrence of axial chromatic aberration can be suppressed effectively.
The optical element according to the present invention may be positioned in the vicinity of an object or an image plane.
With the above-described arrangement, favorable effects can be obtained with respect to astigmatism, coma, image distortion, and curvature of field. When the refractive index is increased, it becomes possible to reduce the occurrence of various aberrations. Consequently, the load on another optical element used to correct these aberrations reduces favorably.
Further, it is desirable to satisfy the following condition:
xcexd greater than xe2x88x92195n+352.5xe2x80x83xe2x80x83(1)
where n is the refractive index for the spectral d-line of the organic-inorganic composite material and xcexd is the Abbe""s number thereof.
FIG. 5 is a diagram showing the relationship between the refractive index n for the spectral d-line and Abbe""s number xcexd of existing plastic materials and organic-inorganic composite materials used in Examples (described later) of the present invention. The refractive index n and Abbe""s number xcexd of each of the materials are as follows (in FIG. 5, ♦ represents the existing plastic materials, xe2x97xaf represents the organic-inorganic composite materials used in Examples).
In FIG. 5, the straight solid line represents the relationship of xcexd=xe2x88x92195n+352.5. Organic-inorganic composite materials located above the solid line [i.e. satisfying condition (1)] exhibit reduced wavelength dispersion and are favorable for use to form optical elements of optical devices using a white light source in particular.
It is more desirable to satisfy the following condition:
xcexd greater than xe2x88x92175n+326xe2x80x83xe2x80x83(2)
where n is the refractive index for the spectral d-line of the organic-inorganic composite material and xcexd is the Abbe""s number thereof.
In FIG. 5, the straight dashed line represents the relationship of xcexd=xe2x88x92175n+326. Organic-inorganic composite materials located above the dashed line [i.e. satisfying condition (2)] exhibit further reduced wavelength dispersion and are even more favorable for use to form optical elements of optical devices using a white light source in particular.
Regarding the Abbe""s number, it is desirable to satisfy the following condition:
20 less than xcexd less than 65xe2x80x83xe2x80x83(3)
If the Abbe""s number is reduced to increase dispersion, the amount of chromatic aberrations produced by the refracting surfaces increases. Because the reflecting surface produces no chromatic aberration in theory, the chromatic aberrations produced in the optical element cannot be corrected by the reflecting surface. The chromatic aberrations of the optical element can be corrected by another optical element when an optical system is constructed by combining together a plurality of optical elements. However, it is preferable that the amount of chromatic aberrations produced by the optical element should be minimal from the beginning because it would be possible to construct an optical system with a reduced number of optical elements. If the Abbe""s number xcexd is not larger than the lower limit of the condition (3), i.e. 20, dispersion becomes excessively large. Consequently, aberrations produced by the optical element become excessively large and hence difficult to correct by another surface. Conversely, if the Abbe""s number xcexd is not smaller than the upper limit, i.e. 65, it becomes difficult to obtain a sufficiently high refractive index. Consequently, aberrations produced by the optical element become excessively large and hence difficult to correct by another surface.
Regarding the refractive index n for the spectral d-line, it is desirable to satisfy the following condition:
1.6 less than n less than 1.9xe2x80x83xe2x80x83(4)
If the refractive index n is not larger than the lower limit of the condition (4), i.e. 1.6, the refractive index becomes excessively small. Consequently, spherical and other aberrations produced by the optical element become excessively large and hence difficult to correct by another surface. If the refractive index n is not smaller than the upper limit, i.e. 1.9, dispersion becomes large. Consequently, chromatic aberrations produced by the optical element become excessively large and hence difficult to correct by another surface.
Still other objects and advantages of the invention will in part be obvious and will in part be apparent from the specification.
The invention accordingly comprises the features of construction, combinations of elements, and arrangement of parts which will be exemplified in the construction hereinafter set forth, and the scope of the invention will be indicated in the claims.